Solve ∫ (from 1 to 2) of ln(3x^2) wrt x=

$$\int _ { 1 } ^ { 2 } \ln ( 3 x ^ { 2 } ) d x =$$

Evaluate

$$\ln(48)-2\approx 1.871201011$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \log_{e}\left(3x^{2}\right)\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{\int \ln(3x^{2})\mathrm{d}x}{\ln(e)}$$

Simplify.

$$\frac{x\left(\ln(3)+\ln(x^{2})-2\right)}{\ln(e)}$$

Simplify.

$$x\left(\ln(3)+\ln(x^{2})-2\right)$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$2\left(\ln(3)+\ln(2^{2})-2\right)-\left(\ln(3)+\ln(1^{2})-2\right)$$

Simplify.

$$-2+\ln(48)$$